A very beautiful classical theory on field extensions of a certain type (Galois extensions) initiated by Galois in the 19th century. Explains, in particular, why it is not possible to solve an equation of degree 5 or more in the same way as we solve quadratic or cubic equations. You will learn to compute Galois groups and (before that) study the properties of various field extensions.
We first shall survey the basic notions and properties of field extensions: algebraic, transcendental, finite field extensions, degree of an extension, algebraic closure, decomposition field of a polynomial.
Then we shall do a bit of commutative algebra (finite algebras over a field, base change via tensor product) and apply this to study the notion of separability in some detail.
After that we shall discuss Galois extensions and Galois correspondence and give many examples (cyclotomic extensions, finite fields, Kummer extensions, Artin-Schreier extensions, etc.).
We shall address the question of solvability of equations by radicals (Abel theorem). We shall also try to explain the relation to representations and to topological coverings.
Finally, we shall briefly discuss extensions of rings (integral elemets, norms, traces, etc.) and explain how to use the reduction modulo primes to compute Galois groups.
PREREQUISITES
A first course in general algebra — groups, rings, fields, modules, ideals. Some knowledge of commutative algebra (prime and maximal ideals — first few pages of any book in commutative algebra) is welcome. For exercises we also shall need some elementary facts about groups and their actions on sets, groups of permutations and, marginally,
the statement of Sylow's theorems.
ASSESSMENTS
A weekly test and two more serious exams in the middle and in the end of the course. For the final result, tests count approximately 30%, first (shorter) exam 30%, final exam 40%.
There will be two non-graded exercise lists (in replacement of the non-existent exercise classes...)
Do you have technical problems? Write to us: coursera@hse.ru

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From the lesson

Week 5

We apply the discussion from the last lecture to the case of field extensions. We show that the separable extensions remain reduced after a base change: the inseparability is responsible for eventual nilpotents. As our next subject, we introduce normal and Galois extensions and prove Artin's theorem on invariants. This week, the first graded assignment is given.

Taught By

Ekaterina Amerik

Professor

Transcript

So now we are ready to give the main definition of this course. So, definition: A Galois extension is a normal and separable extension. So, this will be the central object of Galois theory, Theorem 4: Let L be a finite extension of K, then the number of automorphisms of L over K is less or equal than the degree of L over K with the equality if and only if L is Galois. And equality holds if and only if L is Galois. This is a very easy theorem. What we know, we know that this group of automorphisms of L over K acts freely on the set of homomorphisms. Of K-homomorphisms from L to K-bar. So the number of those automorphisms is equal to the cardinality of an orbit of such action. So, a number of elements in an orbit of this action, which is less or equal than the cardinality of the set itself. And the equality holds whenever this action is transitive. And we have just seen in Theorem 3, that this is the case this means that the extension that L is normal over K. So we have the cardinality of the group is less or equal than the number of homomorphisms. And this is less or equal than the degree of the extension. Here equality means normal, and here equality means separable. So if we look at the comparison between the number of automorphisms and the degree of the extension, then it's less or equal. And equality means Galois. Well, let's make some remarks on normal extensions. So if L is normal over K then, first of all, if we have some isomorphism of subextensions, then it extends to an automorphism of L. to an automorphism of L. Well, to see this, we embed L into an algebraic closure K-bar. And remark that phi extends to a map from L_1 to K-bar, but all those maps have the same image, namely L, right? Well, secondly, the group of automorphisms of L over K acts transitively on the roots of any irreducible polynomial. Again, an isomorphism of stem fields extends to an automorphism of L. And thirdly, if this group fixes some x, which does not belong to K, then x is a purely unseparable element. This is clear because by the previous remark, its minimal polynomial must have x as the only root. has a single root x. has a single root x. So in particular, if L is Galois, if L is separable, then the set of elements which are fixed by the automorphisms of L over K is just K itself. Well, maybe you have not seen this notation. So if G is a group acting on a set X, then X^G is the set of invariants. those x which are not moved by G. Well, definition: if L is Galois, then the Galois group denoted by Galois of L over K is the group of automorphisms of L over K. Let me formulate, before formulating, let us just restate what I just have written here. So L, the invariants of L by the Galois group of L over K, of L by the Galois group of L over K, are the field K.

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